Integrand size = 13, antiderivative size = 16 \[ \int \frac {1}{x^4 \sqrt {-1+x^6}} \, dx=\frac {\sqrt {-1+x^6}}{3 x^3} \]
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Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \int \frac {1}{x^4 \sqrt {-1+x^6}} \, dx=\frac {\sqrt {x^6-1}}{3 x^3} \]
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Rule 270
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-1+x^6}}{3 x^3} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 \sqrt {-1+x^6}} \, dx=\frac {\sqrt {-1+x^6}}{3 x^3} \]
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Time = 0.80 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
trager | \(\frac {\sqrt {x^{6}-1}}{3 x^{3}}\) | \(13\) |
risch | \(\frac {\sqrt {x^{6}-1}}{3 x^{3}}\) | \(13\) |
pseudoelliptic | \(\frac {\sqrt {x^{6}-1}}{3 x^{3}}\) | \(13\) |
gosper | \(\frac {\left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{3 x^{3} \sqrt {x^{6}-1}}\) | \(33\) |
meijerg | \(-\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \sqrt {-x^{6}+1}}{3 \sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, x^{3}}\) | \(33\) |
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none
Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 \sqrt {-1+x^6}} \, dx=\frac {x^{3} + \sqrt {x^{6} - 1}}{3 \, x^{3}} \]
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Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \frac {1}{x^4 \sqrt {-1+x^6}} \, dx=\begin {cases} \frac {i \sqrt {-1 + \frac {1}{x^{6}}}}{3} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\\frac {\sqrt {1 - \frac {1}{x^{6}}}}{3} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^4 \sqrt {-1+x^6}} \, dx=\frac {\sqrt {x^{6} - 1}}{3 \, x^{3}} \]
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none
Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {1}{x^4 \sqrt {-1+x^6}} \, dx=\frac {\sqrt {-\frac {1}{x^{6}} + 1}}{3 \, \mathrm {sgn}\left (x\right )} - \frac {1}{3} \, \mathrm {sgn}\left (x\right ) \]
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Time = 5.15 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^4 \sqrt {-1+x^6}} \, dx=\frac {\sqrt {x^6-1}}{3\,x^3} \]
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